Solvable world problems in semigroups

Cummings, P. A.; Goldstein, Richard Z.

doi:10.1007/bf02573520

Cited by 6 publications

(13 citation statements)

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“…For a published version of this work, see Theorems 5.2.14 and 5.2.15 in Peter Higgins' book [10]. Using Remmers' geometric method, Cummings and Goldstein, [6], proved that the word problem is also solvable for finite semigroup presentations which satisfy the small overlap hypothesis C(2) and also a certain semigroup hypothesis T (4). As the reviewer of their paper points out, they actually prove a conclusion that is strong enough to obtain the same consequence that is noted above for Remmers' result.…”

Section: Typeset By a M S-t E Xmentioning

confidence: 99%

“…Following [6] and [11], we state a definition for the semigroup hypothesis, T (4). Assume for the semigroup presentation, A : R , that the set R is transitively closed: that is, if w 1 = w 2 and w 2 = w 3 are both relations in R, then R also includes the relation w 1 = w 3 .…”

Section: Proofmentioning

confidence: 99%

“…Theorem 3.4, (Cummings and Goldstein, [6]). Suppose that the semigroup S has a finite presentation, A : R , which satisfies the small overlap hypotheses, C(2) and T (4) and that δ is a bound on the lengths of the R-words.…”

Section: Proofmentioning

confidence: 99%

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Decision and Separability Problems for Baumslag–solitar Semigroups

Jackson

2002

Int. J. Algebra Comput.

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Abstract. We show that the semigroups S k, having semigroup presentations a, b : ab k = b a are residually finite and finitely separable. Generally, these semigroups have finite separating images which are finite groups and other finite separating images which are semigroups of order-increasing transformations on a finite partially ordered set. These semigroups thus have vastly different residual and separability properties than the Baumslag-Solitar groups which contain them. 0:Introduction and NotationLet k and be nonnegative integers. Throughout, we will write S k, for the semigroup having semigroup presentation a, b : ab k = b a . We regard these semigroups as being, in some respects, analogous to the groups G k, having group presentations Gp a, b : ab k = b a . The groups G k, are generally referred to as Baumslag-Solitar groups. Since the Baumslag-Solitar groups have one-relator presentations, it has long been known, by a theorem of Magnus, [15], that they have solvable word problems. The Baumslag-Solitar groups are HNN-extensions over the integers, so brief modern proofs that they have solvable word problems follow from Britton's Lemma. By a theorem of Baumslag and Solitar, [3], many of the Baumslag-Solitar groups are non-hopfian and hence are not residually finite [16]. For a more recent account of Baumslag-Solitar groups, see also [2].When k ≥ 1 and ≥ 1, it is known by a theorem of Adjan, [1, Section II, Theorem 3], that the natural homomorphism from S k, to G k, is an embedding, and hence the presentations given for S k, also have solvable word problems. Moreover, [1, Section II, Theorem 1], a left-cancellation law holds in S k, provided that ≥ 1 and a right-cancellation law holds in S k, provided that k ≥ 1. In Section 1, we will give direct explicit solutions of the word problem for the semigroups S k, (even in the fairly simple cases where k and/or are 0). Later, we will outline an alternate proof of the cancellation laws.When A is a set, regarded as an alphabet of letters, we will use A + to denote the free semigroup of nonempty words on A. The free monoid, A * on A is the set of all words on A, including the empty word. We will write 1 for the empty word.

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Section: Typeset By a M S-t E Xmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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Decision and Separability Problems for Baumslag–solitar Semigroups

Jackson

2002

Int. J. Algebra Comput.

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“…In [1] Cummings, the first named author, and Goldstein demonstrated the solvability of the word problem for finite semigroup presentations, with transitively closed sets of defining relations, that satisfy C (2) and T (4). This result immediately followed once it was shown that the number of edges along the right boundary sides of minimal semigroup derivation diagrams, over such presentations, never exceeds twice the number of edges along their left boundary sides.…”

Section: Introductionmentioning

confidence: 84%

“…Following [1], we say S = X | R satisfies T (4) if this coinitial graph contains none of the 4 subgraphs depicted in Fig. 3, where i = j and k = l.…”

Section: Lemma 26mentioning

confidence: 99%

A solvable conjugacy problem for finitely presented semigroups satisfying C(2) and T(4)

Cummings

Jackson

2017

Semigroup Forum

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There are multiple, inequivalent, definitions for conjugacy in semigroups. In Cummings and Jackson (Semigroup Forum 88, 52-66, 2014), we conjectured that, for at least one of these definitions of conjugacy, the conjugacy problem for finitely presented semigroups satisfying C(2) and T (4) is solvable. Here we essentially verify that conjecture. In that 2014 Semigroup Forum publication, we developed geometric methods to solve a conjugacy problem for finitely presented semigroups satisfying C(3). We use those methods again here.

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A solvable conjugacy problem for finitely presented C(3) semigroups

Cummings

Jackson

2013

Semigroup Forum

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Solvable world problems in semigroups

Cited by 6 publications

References 3 publications

Decision and Separability Problems for Baumslag–solitar Semigroups

Decision and Separability Problems for Baumslag–solitar Semigroups

A solvable conjugacy problem for finitely presented semigroups satisfying C(2) and T(4)

A solvable conjugacy problem for finitely presented C(3) semigroups

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